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THE CUSP CONDITIONS FOR MOLECULAR WAVE FUNCTIONS*
bY
Busse l l T Pack acd Joseph 0. Hirschfelder
Abstract ,;??I2
The cusp condi t ions a re derived which descr ibe t h e
behavior of the wave funct ion a t the s i n g u l a r i t l e s of the Coulomb
p o t e n t i a l corresponding t o the coalescence of two or more
p a r t i c l e s . I n t h i s der iva t ion , the wave funct ion i s not
s p h e r i c a l l y averaged; the f ixed-nuclei approximation i s not
requi red ; and the wave funct ion may have nodes a t t he s ingular
points . I n add i t ion t o t h e genera l t reatment , t he cusp
condi t ions for diatomic molecules a r e discussed i n th ree
d i f f e r e n t coordinate systems.
* This r e sea rch was supported p a r t i a l l y by the Nat ional Aeronautics and Space Administration Grant NsG-275-62 and p a r t i a l l y by a National Science Foundation Graduate Fellowship t o Russe l l T Pack.
https://ntrs.nasa.gov/search.jsp?R=19660026442 2020-08-07T11:00:57+00:00Z
. c
I. In t roduct ion
* L e t us consider a s p a t i z l wave funct ion, ~ ( ~ , , & , w * o ~ ~ d ) 9
which i s an exact so lu t ion of t he N-part ic le , n o n - r e l a t i v i s t i c 11
Schrodinger equation,
( H - E ) I = O .
Here, the Hamiltonian, H = T + V, is the sum of the k i n e t i c
energy operator , T, and the p o t e n t i a l energy, V. The p o t e n t i a l
energy i s the sum of Coulombic terms and is thus s ingular where
any two p a r t i c l e s coalesce. Now, Kato has proved t h a t 3 i s
everywhere bounded. Hence, €3 and HS a r e everywhere
bounded, and t h e s i n g u l a r i t i e s of V must be exac t ly cancel led
i n M=(T+v)$ . This cance l l a t ion w i l l occur only i f
has cusps (or nodes) a t the s ingular po in t s of V.
1
W e thus
l -
def ine the cusp condi t ions as desc r ip t ions of t he proper
behavior of 3 a t those s ingular po in ts .
For an N-electron atom i n the heavy-nucleus approximation,
1 Kato proved t h e following cusp condi t ions: %_--
- - - - - * A l l sp in dependence i s omitted u n t i l Sec t ion V I .
Here, i s the d i s t ance between the coa lesc ing p a r t i c l e s 1
and 2 ; i s averaged over a small sphere about t h e
s i n g u l a r i t y ; and 'd i s a cons tan t . A t an e l ec t ron -e l ec t ron
s ingu la r i ty , \(= & , while a t a nucleus-electron s ingu la r i ty ,
$ = -Z. Here Z i s the nuclear charge.
2 Bingel has in t eg ra t ed Kato's r e s u l t and removed the
spher ica l average r e s t r i c t i o n by adding an angular dependent
t e r m t o obta in
rl
where
gtZ depends on the other p a r t i c l e s and i s no t f u l l y spec i f i ed
by the behavior of 3 a t V&=O . E q . (3) i s v a l i d fo r
molecules as w e l l a s atoms, bu t i t is v a l i d only i n the f ixed-
nuc le i approximation, and i t does not apply i f 3 a t fa=O . From Eq. (3) Bingel derived the cusp condi t ions
on the f i r s t order d e n s i t y matrix and the p r o b a b i l i t y dens i ty .
S t e ine r3 has a l s o derived the cusp condi t ions on the p r o b a b i l i t y
dens i ty of an N-electron atom. Hirschfe lder has presented a method
fo r removing e l ec t ron -e l ec t ron s i n g u l a r i t i e s from the Hamiltonian,
bu t it has l i m i t e d a p p l i c a b i l i t y s ince the r e s u l t i n g Hamiltonian
has a node
4
r
i s not Ilernritian.
In our study, the cusp conditions a r e derived d i r e c t l y by I f
solving t h e Schrodinger equation where two p a r t i c l e s a r e very
c
c lose together . The fixed-nuclei approximation and angular
averaging a r e not required, and nodes i n 3 a r e allawed.
As a spec ia l case we discuss the cusp condi t ions i n confocal
e l l i p t i c coordinates f o r f ixed-nuclei diatomic molecules.
Then, we consider the coalescence of three o r m o r e p a r t i c l e s
and genera l ize our two-particle r e s u l t s t o f ind many-particle
cusp condi t ions which a r e s u f f i c i e n t t o insure removal of a l l
s i n g u l a r i t i e s involved. No d i f f i c u l t y r e s u l t s from the
inc lus ion of sp in i n the wave function.
The s a t i s f a c t i o n of t h e cusp condi t ions should imp, rove an
approximate wave function. Indeed, an approximate wave funct ion
must s a t i s f y t h e cusp conditions t o give good values fo r e i t h e r
the charge dens i ty near the Coulomb s i n g u l a r i t i e s or the
f i e l d gradient ( Crc qi3 ). s ingular i t ies i s probably pa r t i cu la r l y important i n per turba t ion
theory . But the s a t i s f a c t i o n of t he cusp condi t ions does not
seem t o g r e a t l y a f f e c t the energy ca lcu la ted by v a r i a t i o n a l
treatments.
and Rayleigh-Ritz energy optimization ca l cu la t ions on Helium
using cor re la ted o r b i t a l s of s eve ra l forms.
which they obtained from t h e i r wave funct ions converged
I n addi t ion, the removal of t he
5
Roothaan and Weiss6 ca r r i ed out Hartree-Fock
The values of y
3
4 ' '
very slowly toward the c o r r e c t va lues h s more genera l o r b i t a l
forms were chosen.
was observed by Kolos and Rar~thaar,' i n t h e i r ca l cu la t ions of
Sircilar slow ccnvergence of t he value of 1
corre la ted o r b i t a l s fo r the H2 rnolecale. Is, fac t , t h e
convergence i s so slow t ha t , as Pekeris8 noted, h i s extremely
accurate, 1078-term Helium wave func t ion s t511 has a 5 per cent
e r r o r i n i t s e lec t ron-e lec t ron vzlue o f However, K i m ,
Chang, and Hirschfelder ' found t h a t imposing the cusp condi t ions
on a Guillemin-Zener wave func t ion fo r Hl d i d not r a i s e the
ca lcu la ted energy s i g n i f i c a n t l y . And Conrojr'O used the cusp
condi t ions t o r e s t r i c t his choice of wave funct ions i n h i s
v a r i a t i o n a l ca l cu la t ions of the energy of a number of small
diatomic molecules.
11. Derivation of the Cusp Conditions
I ! We der ive the cusp condi t ions by expanding the Schrodinger
equation about a s ingular po in t of the po ten t i a l , V . L e t u s
begin by pu t t ing the equat ion i n t o a form which s i m p l i f i e s
the expansion. I n a space-fixed coordinate system the N-part ic le
Schrodinger equat ion i s I 1
5
Eq. ( 4 ) is i n atomic u n i t s : the dis tances , rjj a re i n
u n i t s of the Bohr rad ius , the masses, mi the e l ec t ron mass, and the charges, zL proton charge. Iet us focus a t t e n t i o n on p a r t i c l e s 1 and 2
(which could be any two of the
t o t h e i r cen ter of mass, 6&=(*,& + m Z f i ~ / ( W , + ~ ) , and r e l a t i v e , r , 2 ~ x,-fz , coordinates .
coord ica tes of p a r t i c l e s 3,. -, N unchanged. Then,
, a re i n u n i t s of
, a r e i n u n i t s of t he
N p a r t i c l e s ) by transforming
* We leave the
p a r t i c l e s 1 and 2 wr i t t en e x p l i c i t l y , t he Schr:dinger
equation becomes
- - - - - 11
* I f Eq. ( 4 ) w e r e t he e l ec t ron ic Schrodinger equat ion of an atom or molecule i n the f ixed-nucle i approximation, the coordinates would already be the r e l a t i v e coordinates for nucleus-e l ec t ron s i n g u l a r i t i e s .
b r
6
To consider Eq. (6) near the s ingular point , c;,so , l e t
p a r t i c l e s 1 and 2 be wi th in an a r b i t r a r i l y small d i s tance
of each other (()sr,,& & )
be separated ( 6 4 4 ' r,~
On t h i s i n t e rva l , V ; z = ~ i s the only s ingular po in t of V. I f
we now examine Eq. (6) and r e t a i n e x p l i c i t l y only those p a r t s
which w i l l be of lower order i n qz than 3 , i t i s
c l ea r t ha t
, but l e t a l l other p a r t i c l e s
fo r a l l . t 5 i 6 3 with 34j<N),
where O(em) implies terms of order n and higher i n Yiz . To conveniently expand E , l e t rlz be expressed i n
spher ica l polar coordinates %,and qt ) . Then,
The spher ica l harmonics, Xm(&,qm\, a r e the only bounded
eigenfunctions of . They s a t i s f y the condi t ion, 2
- * 7
9
I
of Eq. (8)- Hence, 'p e m be written as tke sum
When t h i s sum is subs t i t u t ed i n t o Eq. (81, the. Ennesr
independence of the YAW requires that the c c e l f i c i e n t of
i r t he resulting EUZ is zeros
The po in t vuso order d i f f e r e n t i a l equation (10). Therefore, Eq, (SO) must
have a t Peast one so lu t ion of t h e f o r m ,
is a rega lar singular p i n t of t h i s second
21,1%
where uti,* 0 , i n general, It should be remembered
are bounded, continuous functions of the positfans of t he
other p a r t i c l e s . Nsw if we s u b s t i t u t e t h i s sum i n t o Eq. (IO)
and c o l l e c t powers Of 3~ 9
Due t o the l i n e a r independence of the powers of
i n t e r v a l
be zero, and we ge t e x p l i c i t expressions f o r two r ecu r s ion
r e l a t i o n s ,
c2 on the
05 Vi2 - L e , t he c o e f f i c i e n t of each power must
The s a t i s f a c t i o n of t hese two r e l a t i o n s in su res t h a t both
the Tiz and the ve' 12
equation.
But 4(h=-@+i)
s i n g u l a r i t i e s are removed from the
From Eq. (12) , e i t h e r a(&= A or ~(~z-dlfl) must be r e j e c t e d as it produces a s o l u t i o n
which is unbounded a t q2=0 . S e t t i n g e$$A i n
Eq. ( 1 2 ) and Eq. (13), w e ob ta in a s o l u t i o n f o r fim ,
I n f ac t , fo r any al lowable ,( , the func t ion fAm i s the
only bounded s o l u t i o n of Eq. (10). This can be e a s i l y
demonstrated by us ing the Wronskian methodL3 t o cons t ruc t t he
o ther s o l d i o n .
. 9
I .-
Now, l e t ')? be the sm1Zes t value of 1 for which
We*# 0 e Then, as qs30 , +fz . Thus, i f
m>o the function 3 has a male a t $%=O ; and i f
n = 0, the function 3 has a cusp at In e i ther
case the cusp conciitiolis, tfie ~ehsv ior .rif 3 are specif ied by 3 i t se l f , which i s
X%=O
These cusp conditions can a lso be wrltten in a differentiated
farm analogms to Kato's y e s u i t ( E q . (2)),
ma-h t y '?
For the s p e c i a l case of n = 0 , t he cusp condi t ions can .
be w r i t t e n more simply. S e t t i n g , n = 0 i n Eq. (15), we see
t h a t
I f w e a l s o wr i t e the sphe r i ca l harmonics i n t h e i r r e a l forms,
we have
where the funct ions tu,*), (aZ,r3;*oJG) are given by L
U O ; 00
e tc . . The d i f f e r e n t i a t e d form of the cusp condi t ions i s thus
(We note t h a t Z,&#&,z reduces t o the d of Eq. ( 2 ) i n
t he f ixed nuc le i approximation.) Since J;,&e,C*od- %a 9
e t c . , we can a l s o write Eq. (17) i n t e r m s of vec tors . This
g ives
which i s Bingel ' s r e s u l t (Eq. ( 3 ) ) .
111. Diatomic Molecule Cusp Conditions
I n t r e a t i n g diatomic molecules, i t i s convenient t o know
the cusp condi t ions i n add i t iona l coordinate systems.
s ec t ion we present t h e nucleus-electron cusp condi t ions i n
conf oca1 e l l i p t i c coordinates , i n t e r - p a r t i c l e coordinates, and
s p h e r i c a l polar coordinates for the s p e c i a l case of the
N-electron, f ixed-nucle i diatomic molecule.
I n t h i s
14 A . Confocal E l l i p t i c (Pro la te Spheroidal) Coordinates
The cusp condi t ions can be obtained i n d i f f e r e n t coordinate
systems by making a change of va r i ab le s i n Eq. (15).
i n t he case of confocal e l l i p t i c coordinates , it i s simpler
t o de r ive the cusp condi t ions d i r e c t l y .
i s s imi l a r t o t h a t given i n the previous sec t ion , we present
only a summary of t he r e s u l t s .
However,
Because t h e d e r i v a t i o n
II
We f i r s t w r i t s the e l e c t r o n i c Schrodinger equat ion i n
confocal e l l i p t i c coordinates and then expand i t abcut a
nucleus-electron s i n g u l a r i t y of the pcjtential , V. The confocal
e l l i p t i c coordinates of the i t h e l e c t r o n a re fL'c(rb(+ * y R , yi= c'Ctd-Gp)/R , and Td Here 06 a d e a r e the
two nuclei , R i s the in t e rnuc lea r d i s t ance , and 9 i s the
azimuthal angle. The ranges of the va r i ab le s a re : I & TlAd9
2; 4 / , and 0 5 27r I n these coordinates ,
the e l ec t ron ic Schrgdinger equat ion i s
where
Now, l e t e l e c t r o n i be very c l o s e t o nucieus o( (/ff,fl+(
- and -1 68 &- j+L)
separated from a l l o the r p a r t i c l e s
f o r a l l L ) i , and E(<vij f o r a l l i and j 5 but any
R ). Then, i n t h i s region, f,=l, -y,=-l i s the
only s i n g u l a r po in t of V. I f w e w r i t e exp lLc i t ly
, but l e t the o ther e l ec t rons be
( g d < fr-Tr ,
13
only those p a r t s of Eq. (20) whleh w ' l l l be of h w e r order i n e
than , we f i n d t h a t
-2
Eq. ( 2 2 ) i s separable on our reg ion , W e l e t ~=~(~ , )N~~~(h#) and ob ta in the following three equat ions :
Here mz and A a r e separa t ion c m s t a n t s . The s o l u t i o n s of
t he e q c a t i m a r e
where m = 0,1,2, ...,e . The only Sounded nlolutions of the
11 and r, eqLatboas a x obteined by expressing %%A as
a s e r i e s i n (f,-i) and Mmr as a seriea :n 6w) These so lu t ions a r e :
and
I n general , 3 w i l l be a l i n e a r supe rp r - i t i on of such
so lu t ions . I f n 5s the smal les t rn fo r which A, and
are not zero, the so lu t ion and the cusp condi t ions a r e given by *
(29) - - - - - * The so lu t ion fcr elect?crzs 1 'i?es,r nucleus Ls obtained by
simply chaQging a: t o -7 a-nd -@&-
equation. The 1, t e r n are unchanged
t r , -RQ-+) I n t h i s J .I
15
Here A,,,(r,j8 -9%) and &&“..,*b*,f~) a re bounded, continuous
func t ions not spec i f ied by the behsvior of a t the s i n g u l a r i t y .
I f 77>0 i n Eq. (29), has a tode a t ~ ~ ~ , ? & ~ - ) .
But i f n = 0, then 3 has a cusp which i s described by
H e r e
then Z I t he cusp condi t ions i n confocal e l l i p t i c coordinates can
be w r i t t e n i n t h e s i m p l e d i f f e r e n t i a t e d forms,
and
( 3 2 )
?,= -1
Eq. (31) and E q . (32) can be obtzined from Eq. (20) by
simply s e t t i n g
assuming a l l de r iva t ives a r e bormded, and then picking out the
terms with s ingu la r c o e f f i c i e n t s .
Roothaan7 obtained, for t he case zdsq=\ which are equiva isc t t o E¶ . (31) aEd Eq, (32) .
I n s m h a vayy Kolos and
cusp Conditions
B. In te r -par t 2c le C a o r d h a t e s
The cusp condi t ions i n i n t e r - p a r t i c l e coord ina tes are r e a d i l y
obtained from the confocel e l l i p t i c r e s u l t s by a chenge of
var i ab le s . For example, consider an N-electron, f ixed-nucle i
diatomic molecule, and l e t
The (nonorthogonaf) i n t e r - p a r t i c l e d i s tances , d;a ar.d <p , are given by T'= R(SfTb)/z and qp=R(f4-pl)/z . Changing t o
these coordinates i n Eq. (31) and Eq. (32), we ob ta in the cusp
* condi t ions i n the very simple form,
3 c d a t B C. Spher ica l Polar Coordinates
We can e a s i l y give the diatomic nolecule cusp condi t ions
i n sphe r i ca l po lar coord ina tes .
%- A corresponding equat ion a t f' = O i s obtained by interchanging
Consider the example used i n - - - - - 'P
e( and ,@ in. Eq. ( 3 3 ) .
I ,
16
t
17
II .
p a r t B above. Slcce the n u c l e i are fixed, ve take the in t e rnuc lea r
axis t o be the z axis i n Eq. (17) and flnd t h a t
We note t h a t 3 of e l e c t r o n 1 i n o r d e r for E q . (34) t o become a s simple a s Eq. (33).
must be sphe r i ca l ly symmetric i n the coordinates
I V . P robab i l i t y Density Cusp Conditions
The cusp condi t ions can be appl ied t o the e l e c t r o n p r o b a b i l i t y
dens i ty . The sp in l e s s e lectror ; p r o b a b i l i t y densi ty , pi I of
an N-electron atom or molecule i n the f ixed-nuclei approximation,
i s defined by
ere dq=dRdsdzd on the nucleus ( o( ) of i n t e r e s t , To obta in the cusp condi t ions
, w i t h t he coordinate system centered
, w e begia wi th Eq. (E), the cusp condi t ions on 3 . On P W e note t h a t i n these coordinates
appr oxima t i on, Z,v,z = - z4 , where s4 i s the charge
of nucleus o( We now s u b s t i t u t e Eq. (15) i n t o the d e f i n i t i o n
Of f l
rrr= < , and i n t h i s
and use t h a t f a c t tha t Z + Z * = q & Z , fo r any
18
complex number, . Then, t h e cusp condi t ions a r e s p e c i f i e d
by pi i t s e l f ,
a?
where P
I f n = 0,
Thus, a t r, = 0 t he €unct ion
. . We now put E q . (37 ) i n t o the d i f f e r e n t i a t e d form of the cusp
c ond i t i o n s
c
. 19
H e r e & M& conta ins angular terms from the xm , s i m i l a r t o those of Eq. (17), which w i l l not genera l ly be zero
f o r molecules.
However, for a fixed-nucleus atom, 6 can always be chosen
i n v a r i a n t t o invers ion of t h e coordinates (even p a r i t y ) . Then,
s i n c e the p a r i t y of x,,, is (-1)
with even A . Thus, go, i s zero f o r t h i s case,and we have
3 S te ine r ‘s r e s u l t ,
R , f i w i l l con ta in only y~,,,
For molecules, we can e i t h e r use Eq. (38) or t ake t h e
s p h e r i c a l average of Eq. (37),
and obtain, a s Binge12 d i d ,
4
20 ' '
V. Cusp Conditions a t the Coalescence of M P a r t i c l e s
Since the Coulomb p o t e n t i a l i s a two-par t ic le p o t e n t i a l , it
i s c l e a r t h a t i f an N-par t ic le wave func t ion s a t i s f i e s the ?
cusp condi t ions (Eq. (15)) a t the po in t of coalescence of each
pa i r of i t s p a r t i c l e s , i t removes
Schrodinger equation. Hence, i f any set of t he p a r t i c l e s
coalesce, [ H - E ) $
s i n g u l a r i t i e s from the
I 1
w i l l remain bounded.
However, we are not ab le t o determine d i r e c t l y the form
which the exac t so lu t ion w i l l have near the poin t of coalescence
of t h ree or more p a r t i c l e s . Consider M p a r t i c l e s (3L-MCEJ)
c l o s e together ( 0
p a r t i c l e s separated. Then, Eq. ( 4 ) becomes
V' 6 d fo r 14 L C j 4 M) and the other N-M r i
Since there i s no coordinate system i n which t h i s equat ion i s
separable, we a r e not ab le t o solve it d i r e c t l y .
t he re a re var ious forms f o r
condi t ions (Eq. (15)) for each p a i r of the M p a r t i c l e s .
Hence, we are led t o def ine the M-particle cusp condi t ions
( for f l33
p a r t i c l e s coalesce, t he two-par t ic le cusp condi t ions a r e
s a t i s f i e d f a r each p a i r of t he M p a r t i c l e s .
However,
3 which w i l l s a t i s f y the cusp
) a s any condi t ions which insure t h a t when M
.
.
21
One of t he s implest forms for 3 which s a t i s f i e s t h i s
requirement can be expressed as a product wi th each f a c t o r
s a t i s f y i n g a two-par t ic le cusp condi t ion. For example, a poss ib le
form, for a 3 which does no t have a node where the M
4 -e-*: pa1 LLL:es - zoa:esce, woii:?I be
where
and
H e r e 9' i s t h e cen te r of mass of t he M p a r t i c l e s . Direct
s u b s t i t u t i o n of Eq. (41) i n t o Eq. (40) shows t h a t t he
singularities a r e removed.
Genera l iza t ions of cusp condi t ions of t h e form of Eq. (41)
are poss ib le . For example, we can choose 3 and gq t o be 0
4
22 ' -
any bounded, c e r t i ~ z l o u a fdnct ions of a l l the va r i ab le s , f,,**e& 3
such t h a t
and such t h a t
e x i s t and a re bounded f o r a11 k when the M p a r t i c l e s coalesce.
Thus w e a r e ab l e t o cons t ruc t wave funct ions which s u f f i c e
t o remove the s i q g c i s z f t i e s but do not necessa r i ly have %he
same form as the exact sDlution.
V I . Inc lus ion 3f Spin
The N-par t ic le wave func t ion discussed i n the previous
sec t ions i s a s p a t i a l wave func t ion with no sp in dependence. I n
addi t ion , n.o cons idera t ion was given t o t h e Pau l i P r inc ip l e .
I n order t o be phys i ce l l7 acceptable , wave funct ions must be
e igen func t ims G f sZin, praper ly symmetrized with r e s p e c t t o
2,15 interchange o€ p a r t i c l e s . However, it has been shown
t h e exact, properly sysmetrkzed, sp in e igensolu t ions , 9 t h a t
, of
11
t h e Schrodinger equat ion can always be constructed by t ak ing
. I
23
proper l i n e a r combinations of N-part ic le s p a t i a l wave funct ions
times s p i n funct ions. Hence, i f a i l t h e 3 ' s used t o c o n s t r u c t
s a t i s f y t h e cusp conditions, w i l l s a t i s f y t h e cusp
* cond i t ions .
Ac kn ow le d gme n t s
The authors would like t o thank P ro f . W. Byers Brown f o r s e v e r a l
h e l p f u l suggest ions.
- - - - - * The i n t e g r a t e d forms, such as Eq. (15), apply d i r e c t l y i f
t h e Wo;pm c o n t a i n the s p i n funct ions. forms, such a s Eq. ( i 6 ) , do not d i r e c t l y apply s i n c e d i v i s i o n by s p i n func t ions is no t def ined.
But d i f f e r e n t l a t e d
References
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2. W. A. Bingel, Z. f . Naturforschung 88a, 1249 (1963).
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f. n 1 l a L l l A K L U C l , 2. Chem. Phys. 39, 3i.45 (i463j.
5. J. 0. Hirschfelder , W. Byers Brown, and S . Epstein, "Recent
/. 3. C. ' T 7 * - - - L Z - l > - - -
Developments i n Pe r tu rba t ion Theory", i n Advances i n Quantum
Chemistry, (ed., Per-Olov Lowdin, Academic Press , New York, 11
N. Y . , 19%) V O ~ . 1, pp. 352-355.
6. C. C. J. Roothaan and A. W. Weiss, Rev. Mod. Phys. 32, 194 (1960).
7. W. Kolos and C. C. J. Rocthaan, Rev. Mod. Phys. 32, 205 (1960).
8 . C. L. Pekeris , Phys. Rev. - 115, 1216 (1959).
9. S . Kim, T. Y. Chang, and J. 0. Hirschfelder , U. of Wisconsin
T h e o r e t i c a l Chemistry I n s t i t u t e P u b l i c a t i o n 40A (23 March 1965).
10. H. Conroy, J. Chem. Phys. 41, 1327-1351 (1964).
11. E. A. Coddington, An In t roduc t ion t o Ordinary D i f f e r e n t i a l
Equations, (Prect ice-Hal l , Inc. , Englewood C l i f f s , N. J.,
1961) p. 158.
12. P. M. Morse and H. Feshbach, Methods of T h e o r e t i c a l Physics
(McGraw-Hi l l , N e w York, N. Y. , 1953) pp. 530-2.
24
13. Ref. 12, p. 525.
14. W. Byers Brown, Disc. Faraday SOC. - 39, 0000 (1965) has der ived
some cusp cond i t ions i n these coord ina te s f o r R very s m a l l .
H e expressed 3 as t h e s e r i e s ,
. I
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